Theorem lereci

Description: The reciprocal of both sides of 'less than or equal to'. (Contributed by NM, 16-Sep-1999)

	Ref	Expression
Hypotheses	ltplus1.1	|- A e. RR
	prodgt0.2	|- B e. RR
Assertion	lereci	|- ( ( 0 < A /\ 0 < B ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	|- A e. RR
2		prodgt0.2	|- B e. RR
3		lerec	|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )
4	2 3	mpanr1	|- ( ( ( A e. RR /\ 0 < A ) /\ 0 < B ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )
5	1 4	mpanl1	|- ( ( 0 < A /\ 0 < B ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) )